Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-5p^3 - 60p^2 - 135p}{5p^2 + 80p + 315}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-5p(p^2 + 12p + 27)} {5(p^2 + 16p + 63)} $ $ t = -\dfrac{5p}{5} \cdot \dfrac{p^2 + 12p + 27}{p^2 + 16p + 63} $ Simplify: $ t = - p \cdot \dfrac{p^2 + 12p + 27}{p^2 + 16p + 63}$ Next factor the numerator and denominator. $ t = - p \cdot \dfrac{(p + 9)(p + 3)}{(p + 9)(p + 7)}$ Assuming $p \neq -9$ , we can cancel the $p + 9$ $ t = - p \cdot \dfrac{p + 3}{p + 7}$ Therefore: $ t = \dfrac{ -p(p + 3)}{ p + 7 }$, $p \neq -9$